Sign-changing solutions for supercritical elliptic problems in domains with small holes |
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Authors: | E N Dancer Juncheng Wei |
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Institution: | 1.School of Mathematics and Statistics,University of Sydney,Sydney,Australia;2.Department of Mathematics,Chinese University of Hong Kong,Shatin,Hong Kong |
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Abstract: | Let \(\mathcal {D}\) be a bounded, smooth domain in \(\mathbb {R}^N\) , N ≥ 3, \(P\in \mathcal {D}\) . We consider the boundary value problem in \(\Omega = \mathcal {D} \setminus B_\delta(P)\) , $\begin{aligned}\Delta u + |u|^{p-1} u = 0\, \quad in\, \Omega,\\u = 0\quad on\, \partial\Omega,\end{aligned}$ with p supercritical, namely \(p > \frac{N+2}{N-2}\) . Given any positive integer m, we find a sequence \(p_1 < p_2 < p_3 < \cdots , \quad with \lim_{k\to+\infty} p_k = +\infty \), such that if p is given, with p ≠ p j for all j, then for all δ > 0 sufficiently small, this problem has a sign-changing solution which has exactly m + 1 nodal domains. |
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